High order Curl-conforming Hardy space infinite elements for exterior Maxwell problems

TL;DR

This paper introduces high order curl-conforming Hardy space infinite elements for accurately solving Maxwell's equations on unbounded domains, ensuring compatibility with the discrete de Rham complex and demonstrating super-algebraic convergence.

Contribution

It develops a novel construction of prismatic Hardy space infinite elements that fit into the discrete de Rham diagram for Maxwell problems, enabling efficient and accurate simulations.

Findings

Super-algebraic convergence observed in numerical tests.

Effective discretization of wave equations on unbounded domains.

Compatibility with the discrete de Rham complex enhances numerical stability.

Abstract

A construction of prismatic Hardy space infinite elements to discretize wave equations on unbounded domains $\Omega$ in $H^1_{loc}(\Omega)$, $H_{loc}(curl;\Omega)$ and $H_{loc}(div;\Omega)$ is presented. As our motivation is to solve Maxwell's equations we take care that these infinite elements fit into the discrete de Rham diagram, i.e. they span discrete spaces, which together with the exterior derivative form an exact sequence. Resonance as well as scattering problems are considered in the examples. Numerical tests indicate super-algebraic convergence in the number of additional unknowns per degree of freedom on the coupling boundary that are required to realize the Dirichlet to Neumann map.